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DrawMorseLink

PD Presentation:

X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X22,11,5,12 X4,21,1,22 X18,14,19,13 X16,8,17,7 X8,18,9,17 X20,16,21,15 X14,20,15,19

Gauss Code:

{{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -11, 10, -8, 9, -7, 11, -10, 6, -5}}

Jones Polynomial:

- q-11/2 + 3q-9/2 - 7q-7/2 + 12q-5/2 - 18q-3/2 + 20q-1/2 - 22q1/2 + 19q3/2 - 15q5/2 + 10q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-16 - q-14 + 2q-12 + q-10 - 2q-8 + 5q-6 - q-4 + 4q-2 + 4 - q2 + 4q4 - 5q6 + q8 - q10 - 3q12 + 2q14 - q16

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + 2a-3z3 + a-3z5 - 2a-1z-1 - 2a-1z - 3a-1z3 - 3a-1z5 - a-1z7 - 5az - 7az3 - 4az5 - az7 + a3z-1 + 3a3z + 3a3z3 + a3z5

Kauffman Polynomial:

- a-6z4 - 4a-5z5 + 2a-4 - 5a-4z2 + 9a-4z4 - 10a-4z6 - a-3z-1 + 2a-3z - 10a-3z3 + 21a-3z5 - 15a-3z7 + 5a-2 - 10a-2z2 + 2a-2z4 + 18a-2z6 - 14a-2z8 - 2a-1z-1 + 4a-1z - 13a-1z3 + 22a-1z5 + a-1z7 - 8a-1z9 + 3 + 2z2 - 29z4 + 47z6 - 16z8 - 2z10 - 2az + 2az3 - 14az5 + 28az7 - 12az9 - a2 + 12a2z2 - 34a2z4 + 30a2z6 - 5a2z8 - 2a2z10 + a3z-1 - 2a3z - 7a3z5 + 11a3z7 - 4a3z9 + 5a4z2 - 13a4z4 + 11a4z6 - 3a4z8 + 2a5z - 5a5z3 + 4a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     3   j = 8                   7 1   j = 6                 8 3     j = 4               11 7       j = 2             11 8         j = 0           10 12           j = -2         8 10             j = -4       4 10               j = -6     3 8                 j = -8   1 5                   j = -10   2                     j = -12 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 98]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 98]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 9, 3, 10], X[12, 3, 13, 4], X[10, 5, 11, 6], > X[22, 11, 5, 12], X[4, 21, 1, 22], X[18, 14, 19, 13], X[16, 8, 17, 7], > X[8, 18, 9, 17], X[20, 16, 21, 15], X[14, 20, 15, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -11, 10, -8, 9, -7, > 11, -10, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 7 12 18 20 3/2 -q + ---- - ---- + ---- - ---- + ------- - 22 Sqrt[q] + 19 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 15 q + 10 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 2 -10 2 5 -4 4 2 4 6 8 4 + q - q + --- + q - -- + -- - q + -- - q + 4 q - 5 q + q - 12 8 6 2 q q q q 10 12 14 16 > q - 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 98]][a, z]
Out[8]=	3 3 3 1 2 a 2 z 2 z 3 2 z 3 z 3 3 3 ---- - --- + -- + --- - --- - 5 a z + 3 a z + ---- - ---- - 7 a z + 3 a z + 3 a z z 3 a 3 a a z a a 5 5 7 z 3 z 5 3 5 z 7 > -- - ---- - 4 a z + a z - -- - a z 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 98]][a, z]
Out[9]=	3 2 5 2 1 2 a 2 z 4 z 3 5 3 + -- + -- - a - ---- - --- + -- + --- + --- - 2 a z - 2 a z + 2 a z + 4 2 3 a z z 3 a a a a z a 2 2 3 3 2 5 z 10 z 2 2 4 2 10 z 13 z 3 > 2 z - ---- - ----- + 12 a z + 5 a z - ----- - ----- + 2 a z - 4 2 3 a a a a 4 4 4 5 5 5 3 4 z 9 z 2 z 2 4 4 4 4 z 21 z > 5 a z - 29 z - -- + ---- + ---- - 34 a z - 13 a z - ---- + ----- + 6 4 2 5 3 a a a a a 5 6 6 22 z 5 3 5 5 5 6 10 z 18 z 2 6 > ----- - 14 a z - 7 a z + 4 a z + 47 z - ----- + ----- + 30 a z + a 4 2 a a 7 7 8 4 6 15 z z 7 3 7 5 7 8 14 z > 11 a z - ----- + -- + 28 a z + 11 a z - a z - 16 z - ----- - 3 a 2 a a 9 2 8 4 8 8 z 9 3 9 10 2 10 > 5 a z - 3 a z - ---- - 12 a z - 4 a z - 2 z - 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 5 3 8 4 10 12 + 11 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 8 10 10 2 4 4 2 6 2 6 3 > ----- + -- + ---- + 8 q t + 11 q t + 7 q t + 8 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 7 q t + q t + 3 q t + q t

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